Hundreds of immigrants hoping to escape violence in Yemen and join their families in the United States received an official notice last year from the State Department that started with a long-anticipated message: “Your visa is approved.”

But those turned out to be just words.

After the third version of the Trump administration’s travel ban took effect on Dec. 8, their approvals abruptly turned to rejections, in what lawyers, family members and immigration activists said was a display of bad faith.

The rejections of Yemeni citizens, many of them women and children, seeking travel documents from the small United States Embassy in Djibouti, in East Africa, occurred even though the State Department had said it would not revoke any visas when carrying out the president’s proclamation imposing the new restrictions on immigration.

Many of those rejected had been told by the embassy that their visas were approved as early as last July, after being interviewed by the State Department. The approval notices, on a half-size piece of paper in English and Arabic, also said: “We cannot guarantee how long it will take to print it and have your passport ready for pickup. You should check the status of your visa online.”

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Dr. Mohammed Alawadhi, 37, from Little Rock, Ark., said his wife had been approved for a visa on Dec. 6, but after the third travel ban went into effect, she was rejected.CreditVictor J. Blue for The New York Times

Julie Goldberg, a lawyer in Djibouti, said hundreds of people had come to her doorstep since mid-December. She estimated that more than 300 Yemeni nationals had been rejected so far. “There’s no explanation for getting denied — everybody’s getting denied,” she said in a telephone interview from her home there, which serves as a legal clinic. “It’s complete hopelessness. I have women sobbing.”

But the State Department said there was an explanation: However final it may seem, an approval notice is not a guarantee, said an official with the department who spoke anonymously because its employees are not authorized to speak publicly. Such a notice should have been considered provisional, the official said, because at the time, the vetting of the would-be immigrants was not yet complete. The ban’s assurances, the official said, applied only to those holding actual visas.

But the notice — with the case number written in pen to identify the visa applicant — seemed like the last word to the immigrants and those working on their behalf. “We looked at it like that for all of our clients,” said Mosheer Fittahey, a consultant for Marhaba Service in Albany, which assists Yemeni nationals in legal, financial and travel transactions.

Mr. Fittahey said 111 Yemeni-Americans from across the country have contacted him on behalf of their relatives who received approval notices followed by rejections from the embassy in Djibouti. More calls come daily. About 60 percent of the cases there involve women and children, he said.

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The protesters came from Queens and Staten Island and as far away as Buffalo, Michigan and North Carolina.CreditVictor J. Blue for The New York Times

Mohammed Alawadhi, a doctor in Arkansas, said that his wife, Rasha Alzabaidi, 27, who has a heart ailment, received an approval from the embassy in Djibouti on Dec. 6.

On Dec. 4, the Supreme Court lifted two injunctions that had been placed on the second travel ban by lower courts, allowing the State Department on Dec. 8 to carry out the latest ban, which restricts immigration from eight countries, six of them majority Muslim.

On Dec. 17, a day when numerous families were emptying out of the embassy with rejection notices, Ms. Alzabaidi called her husband in a panic, Dr. Alawadhi recalled. “She said to me: ‘Mohammad, they shattered families. I am seeing families in the street crying like they have a death in the family.’”

Four days later, she got her rejection notice. “I was shattered,” Dr. Alawadhi said.

Brooklyn-born Malik Almathil, 27, first applied for a visa for his Yemeni wife, Shaima Almathil, 22, in April 2016. For a year, the couple lived in Malaysia, until she was told to go to Djibouti for her interview in July. She walked out of the United States Embassy in Djibouti on July 31 with her visa approved. As with Ms. Alzabaidi, the approval form told her to check online for when the visa would be printed and she could pick up her passport. She kept checking, to no avail.

On Aug. 14, Samerah Alawdi, 38, a mother of seven whose husband, a naturalized American citizen, cares for four of their United States-born children in Michigan, received the same approval note.

Both waited in Djibouti until she and her three children were also rejected in December.

For people trying to flee Yemen, the rejections have come at considerable cost. With the American Embassy in Sana, the Yemeni capital, closed since February 2015 because of fighting between Houthi rebels and Saudi Arabian-backed military forces, most Yemenis go to Djibouti to apply for their visas. According to relatives, they first have to go through multiple military checkpoints to leave Yemen. Or they must travel on a boat across the Gulf of Aden.

Many say they sold their homes or cars and took out loans in the United States, incurring costs in the thousands of dollars.

The third travel ban restricts immigration from countries that the Trump administration determined had inadequate vetting procedures of prospective immigrants because of faulty information sharing, and had a significant terrorist presence and therefore represented a threat to United States security.

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Julie Goldberg, a lawyer in Djibouti, in East Africa, said that she knew of more than 300 Yemenis who had been rejected after getting notices that their visas were approved.CreditVictor J. Blue for The New York Times

The third ban, issued on Sept. 24, replaced the president’s March executive order and was not supposed to apply to anyone who had a “valid visa” issued before Oct. 18. Case by case waivers were available to those who could show that a denial would cause “undue hardship,” that they would not pose a security threat and that admitting them would be “in the national interest.”

Omar Jadwat, the director of the American Civil Liberties Union’s Immigrants’ Rights Project, one of several groups that sued the government over the travel ban, said the third ban did not seem to take any special cases into account.

“It’s a predictable attempt to maximize the number of people that they ban under this order even when the more logical and humane approach would be to not take away visas that, but for the stamp, were practically already issued,” Mr. Jadwat said.

At the end of December, more than 100 Yemeni-Americans gathered in Foley Square in Lower Manhattan to protest the treatment of their friends and relatives being rejected for visas. They came from Queens and Staten Island, and as far away as Buffalo, Michigan and North Carolina, waving American flags and holding signs like, “We want our Mom,” and “We all have stories to tell.”

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Many of the Yemenis sold their homes or cars and took out loans in the United States, incurring costs in the thousands of dollars, to try and get visas, their relatives said.CreditVictor J. Blue for The New York Times

Lawyers in the United States said that there have also been numerous Iranians overseas with apparent approvals who have been rejected since the latest ban was enacted. As in Yemen, there is no functioning United States Embassy in Iran, forcing people to travel to apply in Saudi Arabia; Armenia; Dubai, in the United Arab Emirates; and Turkey to interview for visas.

Seyed Mousavi, 24, a naturalized citizen from Iran who is an investment banker in Los Angeles, got engaged to his fiancée, Arefé Fayazbakhsh, in July. He said that Ms. Fayazbakhsh, 19, a student at the University of Tehran, had her interview for a visa on Dec. 7 at the American Embassy in Yerevan, Armenia. The consular officer told her that if she stayed in Yerevan for another two days, he could get her the visa.

But on Dec. 8, the day the ban went into effect, while she was still waiting in Armenia, the embassy called her back in and told her she had been rejected. Now the young couple is deciding where they could live, while Mr. Mousavi contemplates giving up his career in the United States.

“It’s really hard for me to go through months and months of uncertainty, with the feeling that I don’t have control of my life, and I have to wait and see what the president of my nation decides for me,” Mr. Mousavi said in a telephone interview recently.

Ms. Goldberg, the lawyer in Djibouti, is planning to file multiple lawsuits in the United States on behalf of her rejected clients, many of them small children.

In the meantime, the embassy in Djibouti is still interviewing visa applicants from Yemen.

Abdo Alfgeeh, 42, who lives in Mohegan Lake, N.Y., applied for his wife and three children, ages 16 to 21, in June 2015. They were finally approved for an interview on Jan. 29.

“They have to be there for the interview,” Mr. Alfgeeh said, “even though there is a big chance they will be refused like everybody else was.”